The linearization of the boundary rigidity problem for MP-systems and generic local boundary rigidity
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911762128830464 |
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| author | Muñoz-Thon, Sebastián |
| author_facet | Muñoz-Thon, Sebastián |
| contents | We consider an $\mathcal{MP}$-system, that is, a compact Riemannian manifold with boundary, endowed with a magnetic field and a potential. On simple $\mathcal{MP}$-systems, we study the $\mathcal{MP}$-ray transform in order to obtain new boundary rigidity results for $\mathcal{MP}$-systems. We show that there is an explicit relation between the $\mathcal{MP}$-ray transform and the magnetic one, which allow us to apply results from [DPSU07] to our case. Regarding rigidity, we show that there exists a generic set $\mathcal{G}^{m}$ of simple $\mathcal{MP}$-systems, which is open and dense, such that any two $\mathcal{MP}$-systems close to an element in it and having the same boundary action function, must be $k$-gauge equivalent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11570 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The linearization of the boundary rigidity problem for MP-systems and generic local boundary rigidity Muñoz-Thon, Sebastián Differential Geometry Analysis of PDEs 53C65, 53C24, 35R30 We consider an $\mathcal{MP}$-system, that is, a compact Riemannian manifold with boundary, endowed with a magnetic field and a potential. On simple $\mathcal{MP}$-systems, we study the $\mathcal{MP}$-ray transform in order to obtain new boundary rigidity results for $\mathcal{MP}$-systems. We show that there is an explicit relation between the $\mathcal{MP}$-ray transform and the magnetic one, which allow us to apply results from [DPSU07] to our case. Regarding rigidity, we show that there exists a generic set $\mathcal{G}^{m}$ of simple $\mathcal{MP}$-systems, which is open and dense, such that any two $\mathcal{MP}$-systems close to an element in it and having the same boundary action function, must be $k$-gauge equivalent. |
| title | The linearization of the boundary rigidity problem for MP-systems and generic local boundary rigidity |
| topic | Differential Geometry Analysis of PDEs 53C65, 53C24, 35R30 |
| url | https://arxiv.org/abs/2401.11570 |